VVV

Vehicle Dynamics
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
–– a
a simulation
simulation tool
tool ––
GERMANY
VEHICLE DYNAMICS
z
ϕ
brh
y
CG
l
β
xv
brv
γ2
Fx1
Fz1
v
γ3
α1
Fy1
x
v1
α3
Fy3
Fy2
α2
Chemnitz
Chemnitz University
University of
of Technology
Technology
Dipl.-Ing.
C. Meißner
Fz2
Fz3
v1
v1Dipl.-Ing.
Dipl.
Dipl.--Ing.
Ing. C.
C. Meißner
Meißner
2008-07-10
2008
2008--07
07--10
10
2008-07-10
1
www.sachsenring.de
1 Introduction
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
Why do we use a simulation of vehicle dynamics?
GERMANY
moving direction
x ICE
VEHICLE DYNAMICS
FD
GB
MD
ICE.... Internal Combustion Engine
GB.... Gearbox (main transmission)
FD.... Front Differential
MD.... Middle Differential
RD.... Rear Differential
drive torque
side forces
We design gearboxes of cars for the following applications:
• main transmission (MT, AT, DCT, CVT)
• final drive (front, rear, middle)
Dipl.-Ing.
C. Meißner
2008-07-10
2
RD
• ancillary aggregates
Torque Vectoring = active drive torque distribution to discrete wheels of vehicles
1 Introduction
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
Why do we use a simulation of vehicle dynamics?
VEHICLE DYNAMICS
GERMANY
Dipl.-Ing.
C. Meißner
2008-07-10
3
1 Introduction
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
Several compoments of the dynamic system:
GERMANY
engine
steering system
....
driver
driver assistants
VEHICLE DYNAMICS
?
tyre modell
clutch (if MT)
gearbox (AT)
road
track
wheel suspension
differential
Dipl.-Ing.
C. Meißner
2008-07-10
4
Sources: www.bmw.de, www.goodyear.de, www.formel1.de, www.fotohomepage.de, www.traction.eaton.com, www.kfz-tech.de, www.m3forum.net, www.bmwm5.com
1 Introduction
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
Demands of tire models
GERMANY
VEHICLE DYNAMICS
Input
Tyre Model
Output
road contact
(surface characteristic,
tyre dimensions,
camber angle, ...)
tyre forces
(x, y, z)
rotatory speed
(centrifugal force, ...)
tyre moments
(Mx, My, Mz aligning
torque)
relative speed
(longitudinal slip,
slip angle, ...)
deformations
(contact area,
damping effects,
power losses, ...)
disturbance
(load changes, ...)
noise
(comfort, ...)
Dipl.-Ing.
C. Meißner
2008-07-10
5
1 Introduction
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
Calculation effort of tyre models
GERMANY
Approximated
measurements
FEM based
Semi-physical models
calculation effort
Dipl.-Ing.
C. Meißner
2008-07-10
6
4000
3000
lateral force [N]
VEHICLE DYNAMICS
experimental effort
2000
1000
source: ATZ
0
.
0
5
10
15
20
slip angle [degrees]
25
30
Content
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
GERMANY
1 Introduction
2 Understanding the behaviour of tyres
3 Simple tyre models
VEHICLE DYNAMICS
3.1 A linear tyre model
3.2 The HSRI tyre model
4 Handling models (more complex)
4.1 The Magic Formula
5 Comfort models (very complex)
5.1 The FTire model
6 Conclusion
Dipl.-Ing.
C. Meißner
2008-07-10
7
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
The nature of wheel deformation
GERMANY
distinguish between longitudinal and lateral direction
VEHICLE DYNAMICS
vertical
direction (z)
zw
xc
Dipl.-Ing.
C. Meißner
2008-07-10
8
longitudinal
direction (x)
δ
xw
α
xw...
α...
δ...
ε...
γ...
yw
moving direction
rolling direction
side slip angle
steering angle
caster angle
camber angle
lateral
direction (y)
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
VEHICLE DYNAMICS
GERMANY
Types of tyres:
radial tyre
bias tyre
General structure of a bias tyre:
Dipl.-Ing.
C. Meißner
2008-07-10
9
source: www.kumho-euro.com
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
Actual tyre research:
GERMANY
„Tweel“:
non-pneumatic wheel
VEHICLE DYNAMICS
from MICHELIN
Dipl.-Ing.
C. Meißner
2008-07-10
10
source: www.michelinman.com, www.fastcoolcars.com
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
GERMANY
The nature of wheel deformation
a) due to a longitudinal force (resp. a torque, x-direction)
example (braked wheel):
rotation
VEHICLE DYNAMICS
belt
Fx
tread element
sliding
undeformed
deformed
deformation
brush model:
Dipl.-Ing.
C. Meißner
2008-07-10
11
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
GERMANY
The nature of wheel deformation
a) due to a longitudinal force (resp. a torque, x-direction)
example (braked wheel):
rdyn... dynamical tyre radius
v.... driving speed
rotation
Dipl.-Ing.
C. Meißner
Fx
Sv = v − rdyn ⋅ ϕɺ
tread element
sliding
undeformed
deformed
deformation
ϕɺ ⋅ rdyn − v
ϕɺ ⋅ rdyn
pure longitudianal force
6
traction force Fx / kN
Relation between
deformation and sliding
v
Sv
deformation
sliding
0
0.2
0.4
0.6
slip κ
0.8
5
4
3
2
1
2008-07-10
12
SV.. relative speed
κ.... longitudinal slip κ =
relativ velocity Sv
VEHICLE DYNAMICS
belt
1
0
0
205/60 R15
Fz=4.0 kN
p=2.0 bar
dry road
0.2
0.4
0.6
0.8
longitudinal slip κ
1
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
GERMANY
The nature of wheel deformation
b) due to a lateral force (y-direction)
carcass
belt
VEHICLE DYNAMICS
Fy
dc
tread element
P
sliding
undeformed
deformed
deformation
Dipl.-Ing.
C. Meißner
2008-07-10
13
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
GERMANY
The nature of wheel deformation
b) due to a lateral force (y-direction)
carcass
ideal deformation:
Fy
2008-07-10
14
sliding
deformation of
the carcass
undeformed
deformed
deformation
α
Trajectory of Point P
sticking
nR
α
U=2·π·R0
sliding
Dipl.-Ing.
C. Meißner
xz wheel plane
tread element
P
dc
dc
sticking
VEHICLE DYNAMICS
belt
Fy
yz
wheel
plane
sliding
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
The phenomenon of the self-aligning torque
GERMANY
the longituninal force acts with a lever arm of nR
x
M z = − Fy ⋅ nR
Fx
VEHICLE DYNAMICS
This torque reduces the slip angle α by
rotating the wheel along the z-axis in the
direction of the velocity.
To keep the intended slip angle, you have
to hold tight the steering wheel.
v
α
y
nR =ny
Fy
nx
with a longitudinal force Fx:
M z = − Fy ⋅ n y − Fx ⋅ nx
Because of slackness in the joints,
a toe-in is required.
δV0
δV0
Dipl.-Ing.
C. Meißner
2008-07-10
15
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
Influencing the self-aligning torque
GERMANY
by moving the reference point
x
VEHICLE DYNAMICS
castor (x-direction)
Fx
z
2008-07-10
16
steering axle
v
α
z
castor angle
ny
y
Fy
nx
x
∆ny
Dipl.-Ing.
C. Meißner
spreading (y-direction)
∆nx
M z = − Fy ⋅ n y − Fx ⋅ nx
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
The interaction between longitudinal and lateral forces
GERMANY
Limit for sticking: resulting force („circle of KAMM“)
Fx
x
sliping
VEHICLE DYNAMICS
Fy
Fx
y
Fx
Fx
sticking
κx‘ κx
κx
Fy_max
Fy
Fy
α‘
Example:
α
• given: longitudinal and lateral forces (Fx, Fy)
α
• if no interaction would be everything could be calculated (κ‘, α‘)
• Fx indicates according to the „circle of KAMM“ the maximum of Fy
• Fy indicates the maximum of Fx
Dipl.-Ing.
C. Meißner
• greater longitudinal slip (κx) and slip angle (α) then without interaction
2008-07-10
17
Remark: The other direction (κx, α → Fx, Fy) is more difficult
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
Typical tyre characteristics:
GERMANY
longitudianal force
lateral force
2008-07-10
18
6
lateral force Fy / kN
0° slip angle
5
3° slip angle
4
3
2
1
0
0
205/60 R15
Fz=4.0 kN
p=2.0 bar
dry road
0.2
0.4
0.6
0.8
longitudinal slip κ
5
0% slip
4% slip
4
3
2
1
0
0
1
self-aligning torque Mz / Nm
Dipl.-Ing.
C. Meißner
traction force Fx / kN
VEHICLE DYNAMICS
6
3
6
9
slip angle α
12
15
self-aligning torque
80
0% slip
60
4% slip
40
20
0
0
3
6
9
slip angle α
12
15
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
The influence of the camber angle
GERMANY
• Influences the tyre deformation
γ
lateral force Fy / kN
z
4
3
2
1
0
100
y
Fz
(-)Fy
γ... camber angle
Dipl.-Ing.
C. Meißner
γ=0°
-6°
+6°
-1
self alignement Mz
VEHICLE DYNAMICS
• negative camber → higher lateral forces
80
60
40
+6°
γ=0°
-6°
20
0
-20
-40
0°
2008-07-10
2°
4°
slip angle α
6°
8°
19
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
The Daimler-Chrysler F400 Carving concept car with active camber variation
VEHICLE DYNAMICS
GERMANY
lever arm for
camber variation
source: www.seriouswheels.com
• active camber variation up to 20°
• special tyres (assymetric, variating
gum consistence)
• 28% higher lateral acceleration
Dipl.-Ing.
C. Meißner
2008-07-10
20
• reduces breaking distance
(5 m with 100 km/h)
source: www.seriouswheels.com
2 Understanding the behaviour of tyres
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
The Influence of the vertical load variation (Fz)
GERMANY
• saturation of lateral forces (compensation by increasing pressure)
lateral force Fy
self-aligning torque Mz
8
2008-07-10
21
8
4°
2°
10°
4
6°
4°
α=2°
0
0 1 2 3 4 5 6
vertical load Fz / kN
7
6
6°
80
40
α=10°
0
0 1 2 3 4 5 6
vertical load Fz / kN
Content
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
GERMANY
1 Introduction
2 Understanding the behaviour of tyres
3 Simple tyre models
VEHICLE DYNAMICS
3.1 A linear tyre model
Dipl.-Ing.
C. Meißner
2008-07-10
22
3.2 The HSRI tyre model
4 Handling models (more complex)
4.1 The Magic Formula
5 Comfort models (very complex)
6 Conclusion
4
2
source: Mitschke. Wallentowitz: Dynamik der Kraftfahrzeuge. Springer Verlag. p. 32
5.1 The FTire model
castor nR / mm
120
self-aligning torque Mz / Nm
6
2
Dipl.-Ing.
C. Meißner
castor nR
160
lateral force Fy / kN
VEHICLE DYNAMICS
• increasing contact area → castor and self-aligning torque increases
7
0
175/70 R13
p=2,1 bar
FzNenn=4,15 kN
γ=0°
α=2°
4°
6°
10°
0 1 2 3 4 5 6
vertical load Fz / kN
7
3.1 A linear tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
longitudianal force
6
0° slip angle
5
lateral force Fy / kN
traction force Fx / kN
GERMANY
VEHICLE DYNAMICS
lateral force
6
c‘κ
F‘x0
4
3
205/60 R15
Fz=4.0 kN
p=2.0 bar
dry road
2
1
0
0
0.2
0.4
0.6
0.8
longitudinal slip κ
0% slip
5
4
3
c‘α
F‘y0
2
1
0
0
1
cκ
3
6
9
slip angle α
12
15
cα
linearized function at the origin:
Fy = cα ⋅ α
Fx = c x ⋅ κ
cx ... periphery stiffness
cα ... cornering stiffness
only for low lateral accelerations (e.g. 4 m/s²)
no interaction between longitudinal and lateral forces
Dipl.-Ing.
C. Meißner
2008-07-10
linearized function at a certain operationg point:
Fy = c'α ⋅ tan(α ) + F ' y 0
Fx = c' x ⋅κ + F ' x 0
23
3.2 The HSRI tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
3.2.1 HSRI tyre model for longitudinal and lateral forces
GERMANY
• HSRI... Higway Safty Research Institute (Michigan)
• extension of the linear tyre model
measurement for the full scale tyre:
Fx
VEHICLE DYNAMICS
Temporary value:
Dipl.-Ing.
C. Meißner
2008-07-10
24
α=0°
Fy
κ=0
sR =
(c x ⋅ κ ) 2 + (cα ⋅ tan(α )) 2
µ ⋅ Fz ⋅ (1 − κ )
cx
cα
κ
cx .... periphery stiffness
cα .... cornering stiffness
Fz ... vertical load
Distinction:
• sR≤0.5
→ sticking
• sR>0.5
→ partial sliding
for smal slip:
Fx = c x ⋅ κ
Fy = cα ⋅ tan(α )
(linear tyre model)
tan(α)
3.2 The HSRI tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
3.2.1 HSRI tyre model for longitudinal and lateral forces
x
Understanding the equation:
GERMANY
sliding
(c x ⋅ κ ) 2 + (cα ⋅ tan(α )) 2
VEHICLE DYNAMICS
sR =
Fres
µ ⋅ Fz ⋅ (1 − κ )
sticking
Fx = c x ⋅ κ
Fmax
Fy = cα ⋅ tan(α )
y
Fres / Fmax = usage ratio of the sticking limit (value range: 0...1)
yes
no
s R ≤ 0.5
stick
partial slip
Fx = c x ⋅ κ
Fx =
c x ⋅ κ s R − 0.25
⋅
1− κ
sR 2
Fy =
cα ⋅ tan(α ) s R − 0.25
⋅
1− κ
sR 2
Dipl.-Ing.
C. Meißner
Fy = cα ⋅ tan(α )
2008-07-10
25
3.2 The HSRI tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
3.2.1 HSRI tyre model for longitudinal and lateral forces
GERMANY
cx
cα
µ
Fz
=
=
=
=
2008-07-10
26
measurements
lateral forces (HSRI)
lateral forces (HSRI)
5
4
4
4
3
2
1
0
lateral force Fy / kN
5
lateral force Fy / kN
5
3
2
1
0
5
10
15
longitudinal slip Sx/%
slip angle a = 0 °
slip angle a = 2 °
slip angle a = 4 °
measurement (a = 0°)
Dipl.-Ing.
C. Meißner
80 000
48 000
1.0
4 000 N
longitudinal forces (HSRI)
longitudinal force Fx / kN
VEHICLE DYNAMICS
Example:
Result:
20
0
3
2
1
0
5
10
slip angle a/°
slip = 0 %
slip = 5 %
slip = 10 %
measurement (Sx = 0%)
15
0
0
2
4
6
vertical load Fz / kN
slip angle a = 2 °
slip angle a = 4 °
slip angle a = 6 °
measurement (a = 4°)
• useable tyre model for small longitudinal slip and small slip angles
• not useable for traction control (TC) and driving with high lateral accelerations
8
3.2 The HSRI tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
3.2.2 HSRI tyre model for self aligning torques (steering response)
GERMANY
The centres of the longitudinal and lateral forces are not in the origin of the coordinate system:
(-)Fy
x
R0
nx
α
z
lL
v
ny
rolling
direction
vertical load Fz
VEHICLE DYNAMICS
Fx
tyre
y
M z = Fx ⋅ nx − Fy ⋅ n y
Fz_stat
c1
s1
lL ≈ 2 ⋅ R0 ⋅ s1
zstat
z
R0... dynamical tyre radius
s1.... tyre sub tangent
Dipl.-Ing.
C. Meißner
2008-07-10
27
3.2 The HSRI tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
3.2.2 HSRI tyre model for self aligning torques (steering response)
GERMANY
yes
VEHICLE DYNAMICS
l
n y = L ⋅ (1 + 2 ⋅ s R ⋅ (0.5 − s R ) )
3
Fy
4
⋅ l L ⋅ tan(α ) +
3
cy
(c x ⋅ κ ) 2 + (cα ⋅ tan(α )) 2
µ ⋅ Fz ⋅ (1 − κ )
lL ≈ 2 ⋅ R0 ⋅ s1
2008-07-10
28
12 − 1 / s R 2  1 − ( s R − 0.5) 
− 1 ⋅ 
n y = lL ⋅ 

C Kor

 
 12 − 3 / s R
 sR − 1 / 3
Fy 
+ 
nx = l L ⋅ tan(α ) ⋅ 
 s R ⋅ ( s R − 1 / 4) c y 
allready known equations:
sR =
Dipl.-Ing.
C. Meißner
no
partial slip
stick
nx =
s R ≤ 0.5
M z = Fx ⋅ nx − Fy ⋅ n y
R0... dynamical tyre radius
s1.... tyre sub tangent
CKor.. correction value
c1.... vertical tyre stiffness
cy..... stiffness of carcas (≈0.5·c1)
3.2 The HSRI tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
3.2.3 HSRI tyre model for vertical load variation (small slip angles)
cornering stiffness
cα / kN/rad
GERMANY
cornering stiffness:
40
30
20
10
0
60
nR = nR 0 ⋅
castor
nR / mm
castor:
Fz
Fz 0
45
30
15
0
FzNenn... reference vertical load
self alignement
coeffizient cMα=cαnR
VEHICLE DYNAMICS

F 
cα =  cα 1 − cα 2 ⋅ z  ⋅ Fz
FzNenn 

50
Dipl.-Ing.
C. Meißner
2008-07-10
50
40
30
20
10
0
0
1
2
3
vertical load Fz / kN
29
3.2 The HSRI tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
3.2.4 HSRI tyre model for camber angles
Dipl.-Ing.
C. Meißner
2008-07-10
30
Fy = cα ⋅ α − cγ ⋅ γ
tan(α) ≈ α for α<<1
M z = cMα ⋅ α + cMγ ⋅ γ
with
cMα = cα ⋅ nR
lateral force Fy / kN
4
1
nR ≈ ⋅ l L
3
3
2
1
0
z
γ=0°
-6°
γ
+6°
-1
100
self-alining torque Mz
VEHICLE DYNAMICS
GERMANY
80
60
40
+6°
γ=0°
y
-6°
Fy
20
0
-20
-40
0°
2°
4°
slip angle α
6°
8°
Fz
4
5
3.2 The HSRI tyre model
CHEMNITZ
UNIVERSITY OF
TECHNOLOGY
3.2.5 Conclusion HSRI tyre model
GERMANY
• koefficients for cornering stiffness (cα) and tangential stiffness (cx)
• temporary value sR with limit for sticking (0.5)
• linear equation while sticking (Fx, Fy)
VEHICLE DYNAMICS
• nonlinear equation while partial sliding (Fx, Fy)
• interaction between lateral and longitudinal forces
extended HSRI Model:
• calculation of the castor (nR, nx, ny, nR0) and the self aligning torque (Mz)
• consideration of camber angle (γ, cγ, cMγγ , cMαα)
• consideration of vertical load variation (s1, c1, Fz0, cα2)
Input parameters:
Dipl.-Ing.
C. Meißner
2008-07-10
31
- 4 parameters for the operating condition (α, κ, γ, Fz) and
- 13 tyre specific parameters (µ, R0, Ckorr, ...)